Hiickel orbital systems

The factors that control if and how these cyclization and rearrangement reactions occur in a concerted manner can be understood from the aromaticity or lack of aromaticity achieved in their cyclic transition states. For a concerted pericyclic reaction to be thermally favorable, the transition state must involve An + 2 participating electrons if it is a Hiickel orbital system, or 4 electrons if it is a Mobius orbital system. A Hiickel transition state is one in which the cyclic array of participating orbitals has no nodes (or an even number) and a Mobius transition state has an odd number of nodes. 

Hiickel-type systems (such as Hilcfcel pericyclic reactions and suprafacial sigmatropic shifts) obey the same rules as for sigma electron. The rationale for this observation is clear If the overlap between adjacent p-electron orbitals is positive along the reaction coordinate, only the peraiutational mechanism can... 

If the transition state resembles the intermediate n-complex, the structure involved is a substituted cyclohexadienyl cation. The electrophile has localized one pair of electrons to form the new a bond. The Hiickel orbitals are those shown for the pentadienyl system in Fig. 10.1. A substituent can stabilize the cation by electron donation. The LUMO is 1/13. This orbital has its highest coefficients at carbons 1, 3, and 5 of the pentadienyl system. These are the positions which are ortho and para to the position occupied by the electrophile. Electron-donor substituents at the 2- and 4-positions will stabilize the system much less because of the nodes at these carbons in the LUMO. 

The tangential pjp orbitals form a Hiickel system for even-membered rings but a Mobius system for odd-membered rings. However, this seems to be of little consequence because it has been shown that both Hiickel and Mobius orbital systems have always an aromatic... 

In the potentially homoantiaromatic molecules of Figure 11, electron delocalization occurs along the periphery of a bicyclic system, involving in this way Aq + 2 rather than 4 q electrons. Since, however, the corresponding orbital system is of Mobius rather than Hiickel type (Figure 9), delocalization of 4q + 2 electrons leads to overall destabilization rather than stabilization. 

It is noteworthy that about the same time as our book was published, two Russian chemists performed Hiickel molecular orbital calculation on C60 (Bochvar Galpern 1973). They named it carbo-s-icosahedrene. Their calculations gave the correct picture of the jt-orbital system. 

Expansion of the Hiickel orbital (HMO) secular determinant for a PAH graph gives the characteristic polynomial P(G X) = det X1 — A where I is the identity matrix and A is the adjacency matrix for the corresponding graph [11]. The characteristic polynomial of a N carbon atom system has the following form... 

Fig. 4.15 The n molecular orbitals and n energy levels for a two-p-orbital system in the simple Hiickel method. The MOs are composed of the basis functions (two p AOs) and the eigenvectors, while the energies of the MOs follow from the eigenvalues (Eq. 4.66). The paired arrows represent a pair of electrons of opposite spin (in the electronic ground state of the neutral ethene molecule i[/ is occupied and i//2 is empty)...

There is, however, an important difference between examples 27 and 41. The later compound forms a Hiickel-aromatic orbital system in 41b while the former compound adopts a Mobius orbital system with 4q + 2 electrons, i.e. 27 is Mobius antiaromatic although six electrons participate in cyclic delocalization (see Section III. B). This is in line with a destabilizing resonance energy of 9.9 kcalmoT (Table 2) calculated with the MM2ERWmethod" "l... 

FIGURE 9. Hiickel and Mobius orbital systems for homoconjugated molecules. In each case, the number of participating electrons (e) is given and classification according to aromatic or antiaromatic character indicated... 

It is important to examine aromaticity in its wicommon features in addition to planarity and aromatic stability. MO calculations carried out by Hiickel in the 1930s showed that aromatic character is associated with planar cyclic molecules that contained 2, 6, 10, 14 (and so on) 7i-electrons. This series of numbers is represented by the term An + 2, where n is an integer, and gave rise to Hiickel s An + 2 rule that refers to the number of 7i-electrons in the p-orbital system. In the case of benzene, n= and thus the system contains six 7C-electrons that are distributed in MOs as shown above. 

First-order perturbation uses the Hiickel orbital coefficients cJfJL of a given parent system to predict the shifts Ssj on the corresponding orbital energies sj. The shift induced by an inductive perturbation bafl is determined by the probability cj that an electron in orbital ifjj resides on atom p. (Equation 4.13). 

One of the key assumptions of the Hiickel approximation is the noninteraction of the TT-orbital system with the a--molecular framework. This is a good approxima-... 

These especially stable molecular orbital systems are found in planar, cyclic, fully conjugated polyenes that contain 4 -I- 2 7t electrons (Hiickel s rule). 

The studies of such closely related structures also ruled out any possible steric effects and the driving influence for reaction rate enhancement has to be seen in the oxygen atom in the y-allylic position (C6 of the Claisen system. Scheme 14). In previous reports [23], Carpenter and Burrows had developed a model to predict the influence of substituents on various pericyclic reactions based on Hiickel orbital energy calculations. According to this approach, a n-... 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

The PEOE method in conjunction with a modified Hiickel Molecular Orbital (HMO) method allows charge calculation in conjugated r-systems. 

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